Discrete Curvature

Theory and Applications


A colloquium on discrete curvature

On a surface theory for quadrilateral nets

Tim Hoffmann

Slides can be found by following the link.

I will report on recent work on a discrete version of surface theory for quadrilateral nets. Our approach aims to generalize the known integrable cases into a more general framework.

There are many well working examples of integrable discretizations of special surface classes as well as well working discrete definitions of fundamental forms, curvatures, shape operator and similar fundamental objects of surface theory for but so far little effort has been made to formulate a general framework that covers the integrable cases with their fundamental properties and still works on a broader class of nets.

This is joint work with Andrew O. Sageman-Furnas (Furnas) and Max Wardetzky.

One of the roots of discrete differential geometry is the theory of discrete integrable surfaces. While in the smooth case these surfaces are often defined by their curvature properties the discretizations originally where done algebraically and only afterwards ad-hoc definition for curvatures where found. The discrete Steiner formula changed that to some extend unifying a notion of curvature for a wider class of nets but it still relies on analogs of special parametrizations. After giving an overview over these developments I will report on recent research that intends to generalize and unify even more discrete surfaces and a broader class of quadrilateral nets.

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